Even and odd functions: the 2 symmetries to know

Definitions

Let $f$ be defined on a domain symmetric about 0 (e.g., $\mathbb{R}$, $[-a, a]$).

• Even: $f(-x) = f(x)$ for all $x$.
• Odd: $f(-x) = -f(x)$ for all $x$.

⚠️ A function need not be even or odd.

Graphical interpretation

• Even: graph symmetric about the y-axis.
• Odd: graph symmetric about the origin (180° rotation).

Reference functions

Function	Parity
$x^2, x^4, x^6, \ldots$	Even
$x, x^3, x^5, \ldots$	Odd
$\cos(x)$	Even
$\sin(x)$	Odd
$\tan(x)$	Odd
$	x
$e^x$	Neither
$\ln(x)$	Domain not symmetric

Method

1. Check domain is symmetric (else impossible).
2. Compute $f(-x)$.
3. Compare with $f(x)$ or $-f(x)$.

Example: $f(x) = 3x^2 - 5$

$f(-x) = 3(-x)^2 - 5 = 3x^2 - 5 = f(x)$ ⇒ even.

Example: $f(x) = x^3 - x$

$f(-x) = -x^3 + x = -(x^3 - x) = -f(x)$ ⇒ odd.

Sum and product

• Even + even = even.
• Odd + odd = odd.
• Even × even = even.
• Odd × odd = even.
• Even × odd = odd.

Application to integrals

• $f$ odd ⇒ $\int_{-a}^{a} f(x)\, dx = 0$.
• $f$ even ⇒ $\int_{-a}^{a} f(x)\, dx = 2 \int_0^a f(x)\, dx$.

Common mistakes

1. Writing $f(-x) = -f(x)$ for “even” (that’s odd).
2. Forgetting to check symmetric domain.
3. Thinking every function is even OR odd ($e^x$ is neither).

?À toi de jouer

f(x) = x⁴ + 2x². Parity?

A.EvenB.OddC.NeitherD.Cannot say

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